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| Mirrors > Home > MPE Home > Th. List > tgbtwntriv2 | Structured version Visualization version GIF version | ||
| Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.) |
| Ref | Expression |
|---|---|
| tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
| tkgeom.d | ⊢ − = (dist‘𝐺) |
| tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
| tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| tgbtwntriv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| tgbtwntriv2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
| Ref | Expression |
|---|---|
| tgbtwntriv2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simprl 771 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥)) | |
| 2 | tkgeom.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
| 3 | tkgeom.d | . . . . . 6 ⊢ − = (dist‘𝐺) | |
| 4 | tkgeom.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
| 5 | tkgeom.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 6 | 5 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐺 ∈ TarskiG) |
| 7 | tgbtwntriv2.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 8 | 7 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 ∈ 𝑃) |
| 9 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝑥 ∈ 𝑃) | |
| 10 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → (𝐵 − 𝑥) = (𝐵 − 𝐵)) | |
| 11 | 2, 3, 4, 6, 8, 9, 8, 10 | axtgcgrid 28471 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵)) → 𝐵 = 𝑥) |
| 12 | 11 | adantrl 716 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 = 𝑥) |
| 13 | 12 | oveq2d 7447 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥)) |
| 14 | 1, 13 | eleqtrrd 2844 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵)) |
| 15 | tgbtwntriv2.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 16 | 2, 3, 4, 5, 15, 7, 7, 7 | axtgsegcon 28472 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 − 𝑥) = (𝐵 − 𝐵))) |
| 17 | 14, 16 | r19.29a 3162 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-trkgc 28456 df-trkgcb 28458 df-trkg 28461 |
| This theorem is referenced by: tgbtwncom 28496 tgbtwntriv1 28499 tgcolg 28562 legid 28595 hlid 28617 lnhl 28623 tglinerflx2 28642 mirreu3 28662 mirconn 28686 symquadlem 28697 outpasch 28763 hlpasch 28764 |
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